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ABSTRACT. A new proof of the irrationality of the number ~(3) is proposed. A new decomposition of this number into a continued fraction is found. Recurrence relations are proved for some sequences of Meyer's G-functions that define a sequence of rationxl approximations to ~(3) =~t the point 1.In 1978 R. Apery proved the following assertion.

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We have studied resonance collisions amongst atoms dressed by a strong laser field. The equations of motion have been solved numerically to yield collisional transition rates for various values of the Rabi frequency ( V) due to the laser and its detuning (A) from the atomic transition frequency (o,). The atoms are assumed to follow classical rectilinear paths and only undergo binary collisions. The resonance interaction is taken to he the first-order dipole-dipole interaction acting within the ground and resonance levels of a pair of atoms. A qualitative explanation is offered for the results obtained by splitting the Hamiltonian into parts that represent the collision-free evolution of the system at the Rahi frequency, the 'normal' resonance interaction, and the 'dressed interaction' responsible for opening new scattering channels. The latter interaction depends upon the relative strengths of the resonance interaction ( R ) , the generalized Rabi frequency (0) and the ratio V I A .